Geodesic
Bounds on Multiple Self-avoiding Polygons
Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 518-530

Voir la notice de l'article provenant de la source Cambridge University Press

A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problemto this study, we considermultiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number ${{p}_{m\times n}}$ of distinct multiple self-avoiding polygons in the $m\,\times \,n$ rectangular grid on the square lattice. For $m\,=\,2,\,{{p}_{2\times n}}\,=\,{{2}^{n-1}}\,-1$ . And for integers $m,\,n\,\ge \,3$ , $${{2}^{m+n-3}}\left( \frac{17}{10} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}\,\le \,{{p}_{m\times n}}\,\le \,{{2}^{m+n-3}}\left( \frac{31}{16} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}.$$
DOI : 10.4153/CMB-2017-072-x
Mots-clés : 57M25, 82B20, 82B41, 82D60, ring polymer, self-avoiding polygon 
Hong, Kyungpyo; Oh, Seungsang. Bounds on Multiple Self-avoiding Polygons. Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 518-530. doi: 10.4153/CMB-2017-072-x
@article{10_4153_CMB_2017_072_x,
     author = {Hong, Kyungpyo and Oh, Seungsang},
     title = {Bounds on {Multiple} {Self-avoiding} {Polygons}},
     journal = {Canadian mathematical bulletin},
     pages = {518--530},
     year = {2018},
     volume = {61},
     number = {3},
     doi = {10.4153/CMB-2017-072-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-072-x/}
}
TY  - JOUR
AU  - Hong, Kyungpyo
AU  - Oh, Seungsang
TI  - Bounds on Multiple Self-avoiding Polygons
JO  - Canadian mathematical bulletin
PY  - 2018
SP  - 518
EP  - 530
VL  - 61
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-072-x/
DO  - 10.4153/CMB-2017-072-x
ID  - 10_4153_CMB_2017_072_x
ER  - 
%0 Journal Article
%A Hong, Kyungpyo
%A Oh, Seungsang
%T Bounds on Multiple Self-avoiding Polygons
%J Canadian mathematical bulletin
%D 2018
%P 518-530
%V 61
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-072-x/
%R 10.4153/CMB-2017-072-x
%F 10_4153_CMB_2017_072_x

[1] [1] Bousquet-Melou, M., Guttmann, A., and Jensen, I., Self-avoiding walks crossing a Square. J. Phys. A 38(2005), no. 42, 9159–9181. http://dx.doi.org/10.1088/0305-4470/38/42/001 Google Scholar

[2] [2] Flory, P., The configuration of real polymer chains. J. Chem. Phys. 17(1949), 303–310. Google Scholar

[3] [3] Guttmann, A., ed., Polygons, polyominos, and polycubes. Lecture Notes in Physics, 775, Springer, Dordrecht, 2009. http://dx.doi.org/10.1007/978-1-4020-9927-4 Google Scholar

[4] [4] Hammersley, J. M., The number of polygons on a lattice. Proc. Cambridge Philos. Soc. 57(1961), 516–523. Google Scholar

[5] [5] Hong, K., Lee, H., Lee, H. J., and Oh, S., Upper bound on the total number ofknot n-mosaics. J. Knot Theory Ramifications 23(2014), no. 13, 1450065. http://dx.doi.Org/10.1142/S0218216514500655 Google Scholar

[6] [6] Hong, K., Lee, H., Lee, H. J., and Oh, S., Small knot mosaics andpartition matrices. J. Phys. A 47(2014), no. 43, 435201. http://dx.doi.org/10.1088/1751-8113/47/43/435201 Google Scholar

[7] [7] van Rensburg, E. Janse, Thoughts on lattice knot statistics. J. Math. Chem. 45(2009), no. 1, 7–38. http://dx.doi.org/10.1007/s10910-008-9364-9 Google Scholar

[8] [8] Lee, H. J., Hong, K., Lee, H., and Oh, S., Mosaic number of knots. J. Knot Theory Ramifications 23(2014), no. 13, 1450069. http://dx.doi.org/10.1142/S0218216514500692 Google Scholar

[9] [9] Lomonaco, S. and Kauffman, L., Quantum knots and mosaics. Quantum Inf. Process. 7(2008), no. 2-3, 85–115. http://dx.doi.org/10.1007/s11128-008-0076-7 Google Scholar

[10] [10] Madras, N. and Slade, G., The Self-avoiding walk. Probability and its applications. Birkhäuser Boston, Boston, MA, 1993. Google Scholar

[11] [11] Oh, S., Hong, K., Lee, H., and Lee, H. J., Quantum knots and the number ofknot mosaics. Quantum Inf. Process. 14(2015), no. 3, 801–811. http://dx.doi.org/10.1007/s11128-014-0895-7 Google Scholar

Cité par Sources :